## Abstract

Let f be a drawing in the Euclidean plane of a graph G, which is understood to be a 1-dimensional simplicial complex. We assume that every edge of G is drawn by f as a curve of constant algebraic complexity, and the ratio of the length of the longest simple path to the the length of the shortest edge is poly(n). In the drawing f, a path P of G, or its image in the drawing π = f(P), is β-stretch if π is a simple (non-self-intersecting) curve, and for every pair of distinct points p ∈ P and q ∈ P, the length of the sub-curve of π connecting f(p) with f(q) is at most βkf(p) − f(q)k, where k.k denotes the Euclidean distance. We introduce and study the β-stretch Path Problem (βSP for short), in which we are given a pair of vertices s and t of G, and we are to decide whether in the given drawing of G there exists a β-stretch path P connecting s and t. The βSP also asks that we output P if it exists. The βSP quantifies a notion of “near straightness” for paths in a graph G, motivated by gerrymandering regions in a map, where edges of G represent natural geographical/political boundaries that may be chosen to bound election districts. The notion of a β-stretch path naturally extends to cycles, and the extension gives a measure of how gerrymandered a district is. Furthermore, we show that the extension is closely related to several studied measures of local fatness of geometric shapes. We prove that βSP is strongly NP-complete. We complement this result by giving a quasi-polynomial time algorithm, that for a given ε > 0, β ∈ O(poly(log |V (G)|)), and s, t ∈ V (G), outputs a β-stretch path between s and t, if a (1 − ε)β-stretch path between s and t exists in the drawing.

Original language | English (US) |
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Title of host publication | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 |

Editors | Susanne Albers |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771504 |

DOIs | |

State | Published - Jun 1 2020 |

Event | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 - Torshavn, Faroe Islands Duration: Jun 22 2020 → Jun 24 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 162 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020 |
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Country/Territory | Faroe Islands |

City | Torshavn |

Period | 6/22/20 → 6/24/20 |

## Keywords

- Dilation
- Geometric spanners
- stretch factor

## ASJC Scopus subject areas

- Software